Properties

Label 2475.4.a.bv
Level $2475$
Weight $4$
Character orbit 2475.a
Self dual yes
Analytic conductor $146.030$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2475,4,Mod(1,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2475.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2475.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(146.029727264\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4 x^{9} - 49 x^{8} + 180 x^{7} + 753 x^{6} - 2420 x^{5} - 4059 x^{4} + 9796 x^{3} + 7226 x^{2} - 9120 x - 2112 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1 + 3) q^{4} + ( - \beta_{4} - \beta_1 + 1) q^{7} + ( - \beta_{3} - \beta_{2} - 4 \beta_1 - 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1 + 3) q^{4} + ( - \beta_{4} - \beta_1 + 1) q^{7} + ( - \beta_{3} - \beta_{2} - 4 \beta_1 - 3) q^{8} - 11 q^{11} + ( - \beta_{8} - \beta_{6} - \beta_1 + 3) q^{13} + (\beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + \beta_1 + 6) q^{14} + (\beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 6 \beta_1 + 16) q^{16} + ( - \beta_{9} + \beta_{6} - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 - 14) q^{17} + (\beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 7 \beta_1 - 14) q^{19} + 11 \beta_1 q^{22} + ( - 2 \beta_{8} - \beta_{7} + \beta_{4} + 2 \beta_{3} - 5 \beta_{2} - 8 \beta_1 + 6) q^{23} + ( - 2 \beta_{9} + 2 \beta_{7} + \beta_{6} + 2 \beta_{5} + 5 \beta_{4} - \beta_{3} + \cdots + 10) q^{26}+ \cdots + ( - 3 \beta_{9} + 2 \beta_{8} - 6 \beta_{7} + 2 \beta_{6} - 15 \beta_{5} + \cdots - 225) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 4 q^{2} + 34 q^{4} + 2 q^{7} - 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 4 q^{2} + 34 q^{4} + 2 q^{7} - 48 q^{8} - 110 q^{11} + 26 q^{13} + 72 q^{14} + 206 q^{16} - 148 q^{17} - 114 q^{19} + 44 q^{22} + 34 q^{23} + 100 q^{26} - 86 q^{28} - 38 q^{29} + 232 q^{31} - 448 q^{32} - 20 q^{34} + 754 q^{37} - 780 q^{38} + 160 q^{41} - 66 q^{43} - 374 q^{44} + 682 q^{46} - 450 q^{47} + 590 q^{49} + 200 q^{52} - 1068 q^{53} + 268 q^{56} - 138 q^{58} - 838 q^{59} - 566 q^{61} - 1230 q^{62} + 462 q^{64} + 430 q^{67} - 2234 q^{68} + 518 q^{71} - 184 q^{73} + 402 q^{74} + 386 q^{76} - 22 q^{77} + 956 q^{79} - 2180 q^{82} - 2094 q^{83} + 892 q^{86} + 528 q^{88} - 512 q^{89} - 858 q^{91} - 4476 q^{92} - 294 q^{94} - 1006 q^{97} - 2226 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4 x^{9} - 49 x^{8} + 180 x^{7} + 753 x^{6} - 2420 x^{5} - 4059 x^{4} + 9796 x^{3} + 7226 x^{2} - 9120 x - 2112 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 19\nu + 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 7 \nu^{9} + 43 \nu^{8} + 348 \nu^{7} - 2064 \nu^{6} - 5511 \nu^{5} + 29915 \nu^{4} + 29298 \nu^{3} - 132422 \nu^{2} - 31632 \nu + 106560 ) / 3264 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3 \nu^{9} + 67 \nu^{8} - 84 \nu^{7} - 3624 \nu^{6} + 7605 \nu^{5} + 63675 \nu^{4} - 99566 \nu^{3} - 368678 \nu^{2} + 224832 \nu + 305856 ) / 6528 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19 \nu^{9} + 107 \nu^{8} + 828 \nu^{7} - 4728 \nu^{6} - 10315 \nu^{5} + 59603 \nu^{4} + 36178 \nu^{3} - 189334 \nu^{2} - 56832 \nu + 52672 ) / 2176 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3 \nu^{9} - 19 \nu^{8} - 108 \nu^{7} + 840 \nu^{6} + 555 \nu^{5} - 10731 \nu^{4} + 8846 \nu^{3} + 36230 \nu^{2} - 40704 \nu - 15552 ) / 384 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 49 \nu^{9} - 165 \nu^{8} - 2436 \nu^{7} + 7104 \nu^{6} + 37761 \nu^{5} - 86597 \nu^{4} - 197198 \nu^{3} + 251034 \nu^{2} + 268752 \nu - 31104 ) / 3264 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 31 \nu^{9} + 103 \nu^{8} + 1580 \nu^{7} - 4536 \nu^{6} - 26135 \nu^{5} + 59983 \nu^{4} + 160698 \nu^{3} - 242574 \nu^{2} - 292288 \nu + 181568 ) / 2176 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 20\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 2\beta_{3} + 28\beta_{2} + 30\beta _1 + 216 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{7} - 4\beta_{6} - 3\beta_{5} + 6\beta_{4} + 36\beta_{3} + 45\beta_{2} + 468\beta _1 + 127 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 40 \beta_{9} + 42 \beta_{8} - 44 \beta_{7} - 40 \beta_{6} - 52 \beta_{5} + 42 \beta_{4} + 92 \beta_{3} + 759 \beta_{2} + 911 \beta _1 + 4927 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10 \beta_{9} + 14 \beta_{8} - 140 \beta_{7} - 186 \beta_{6} - 164 \beta_{5} + 314 \beta_{4} + 1105 \beta_{3} + 1645 \beta_{2} + 11824 \beta _1 + 4819 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1257 \beta_{9} + 1389 \beta_{8} - 1491 \beta_{7} - 1281 \beta_{6} - 1923 \beta_{5} + 1569 \beta_{4} + 3320 \beta_{3} + 20884 \beta_{2} + 28374 \beta _1 + 121012 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 698 \beta_{9} + 1118 \beta_{8} - 5057 \beta_{7} - 6446 \beta_{6} - 6545 \beta_{5} + 11948 \beta_{4} + 32592 \beta_{3} + 55771 \beta_{2} + 313534 \beta _1 + 172965 \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
5.47062
4.33188
3.67029
2.12679
0.983609
−0.207736
−1.29917
−2.21965
−3.94490
−4.91174
−5.47062 0 21.9277 0 0 −16.1408 −76.1933 0 0
1.2 −4.33188 0 10.7652 0 0 30.8635 −11.9784 0 0
1.3 −3.67029 0 5.47106 0 0 −21.0437 9.28196 0 0
1.4 −2.12679 0 −3.47675 0 0 3.88078 24.4087 0 0
1.5 −0.983609 0 −7.03251 0 0 1.05477 14.7861 0 0
1.6 0.207736 0 −7.95685 0 0 −31.6389 −3.31481 0 0
1.7 1.29917 0 −6.31217 0 0 16.4390 −18.5939 0 0
1.8 2.21965 0 −3.07315 0 0 27.0240 −24.5785 0 0
1.9 3.94490 0 7.56227 0 0 −16.6840 −1.72681 0 0
1.10 4.91174 0 16.1252 0 0 8.24545 39.9090 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(11\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2475.4.a.bv yes 10
3.b odd 2 1 2475.4.a.by yes 10
5.b even 2 1 2475.4.a.bx yes 10
15.d odd 2 1 2475.4.a.bu 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2475.4.a.bu 10 15.d odd 2 1
2475.4.a.bv yes 10 1.a even 1 1 trivial
2475.4.a.bx yes 10 5.b even 2 1
2475.4.a.by yes 10 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2475))\):

\( T_{2}^{10} + 4 T_{2}^{9} - 49 T_{2}^{8} - 180 T_{2}^{7} + 753 T_{2}^{6} + 2420 T_{2}^{5} - 4059 T_{2}^{4} - 9796 T_{2}^{3} + 7226 T_{2}^{2} + 9120 T_{2} - 2112 \) Copy content Toggle raw display
\( T_{7}^{10} - 2 T_{7}^{9} - 2008 T_{7}^{8} + 2566 T_{7}^{7} + 1303414 T_{7}^{6} - 777030 T_{7}^{5} - 315738468 T_{7}^{4} + 448111154 T_{7}^{3} + 24120251825 T_{7}^{2} - 104234003248 T_{7} + 82972204116 \) Copy content Toggle raw display
\( T_{29}^{10} + 38 T_{29}^{9} - 111051 T_{29}^{8} - 6346380 T_{29}^{7} + 3208777688 T_{29}^{6} + 265003141424 T_{29}^{5} - 13572774715088 T_{29}^{4} + \cdots + 62\!\cdots\!96 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 4 T^{9} - 49 T^{8} + \cdots - 2112 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - 2 T^{9} + \cdots + 82972204116 \) Copy content Toggle raw display
$11$ \( (T + 11)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} - 26 T^{9} + \cdots + 732524456768 \) Copy content Toggle raw display
$17$ \( T^{10} + 148 T^{9} + \cdots + 95\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{10} + 114 T^{9} + \cdots + 14\!\cdots\!53 \) Copy content Toggle raw display
$23$ \( T^{10} - 34 T^{9} + \cdots + 68\!\cdots\!68 \) Copy content Toggle raw display
$29$ \( T^{10} + 38 T^{9} + \cdots + 62\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( T^{10} - 232 T^{9} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{10} - 754 T^{9} + \cdots - 33\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{10} - 160 T^{9} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{10} + 66 T^{9} + \cdots - 45\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{10} + 450 T^{9} + \cdots - 13\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{10} + 1068 T^{9} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + 838 T^{9} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{10} + 566 T^{9} + \cdots - 10\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{10} - 430 T^{9} + \cdots - 35\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T^{10} - 518 T^{9} + \cdots - 58\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{10} + 184 T^{9} + \cdots - 89\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{10} - 956 T^{9} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + 2094 T^{9} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{10} + 512 T^{9} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + 1006 T^{9} + \cdots - 35\!\cdots\!19 \) Copy content Toggle raw display
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